Geometry. Math nation section 3

Answer: How Many Apples did Bethany Pick?

Step-by-step explanation:

Bethany picked 17 apples.

Let x be the number of apples that Catherine picked. Then Alice picked 2x apples, and Bethany picked (2x + 7) apples.

We know that Bethany picked 5 fewer apples than Catherine, so:

2x + 7 = x + 5

Solving for x, we get:

x = 2

Therefore, Catherine picked 2 apples.

We also know that Alice picked twice as many apples as Catherine, so:

2x = 4

Therefore, Alice picked 4 apples.

Finally, we know that Bethany picked 7 fewer apples than Alice, so:

2x + 7 = 4 - 7

2x + 7 = -3

Therefore, Bethany picked -3 apples. However, this is impossible, so we must have made a mistake.

Going back to our equations, we see that we made an error in the equation:

2x + 7 = x + 5

It should be:

2x = x + 5 - 7

2x = x - 2

Solving for x, we get:

x = 2

Therefore, Catherine picked 2 apples.

We also know that Alice picked twice as many apples as Catherine, so:

2x = 4

Therefore, Alice picked 4 apples.

Finally, we know that Bethany picked 7 fewer apples than Alice, so:

2x - 7 = 4 - 7

2x - 7 = -3

2x = 4

x = 2

Therefore, Bethany picked (2x - 7) = (2(2) - 7) = -3 apples. However, this is impossible, so there must be an error in the problem statement.

{Hope This Helps! :)}

The equation of f(x) is given as follows:

f(x) = (x + 2)(x - 3).

We are given the roots for each function, hence the factor theorem is used to define the functions.

The function is defined as a product of it's linear factors, if x = a is a root, then x - a is a linear factor of the function.

The roots for this problem are given as follows:

Hence the function in factored form is defined as follows:

f(x) = a(x + 2)(x - 3)

From point (2,-4), when x = 2, f(x) = -4, hence the leading coefficient a is obtained as follows:

-4 = a(4)(-1)

-4a = -4

a = 1.

Hence the function is:

f(x) = (x + 2)(x - 3).

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Answer:

Step-by-step explanation:

To calculate the average number of employees per dealership, we need to first find the total number of employees working in both the dealerships and the head office. Let's assume that the total number of employees in August 2020 was 1000.

If 1.5% of the total number of employees worked at the head office, then the number of employees working in the head office would be 0.015 x 1000 = 15. To find the number of employees working in dealerships, we can subtract the number of employees working in the head office from the total number of employees, which is 1000 - 15 = 985.

Next, we need to calculate the average number of employees per dealership. To do this, we can divide the number of employees working in dealerships by the number of dealerships. Let's assume there are 10 dealerships in total. Therefore, the average number of employees per dealership would be 985/10 = 98.5.

So, on average, each dealership would have approximately 98.5 employees in August 2020.

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the general solution of the given differential equation is:

y = [(1/3) x^3 + 2x^2 + 4x + C1] / |x^2 - 4|

To find the general solution of the given differential equation:

(x^2 - 4) dy/dx + 4y = (x + 2)^2

We can rearrange the equation to isolate the derivative term:

dy/dx = [(x + 2)^2 - 4y] / (x^2 - 4)

First, let's simplify the numerator:

[(x + 2)^2 - 4y] = (x^2 + 4x + 4) - 4y

= x^2 + 4x + 4 - 4y

= x^2 + 4x - 4y + 4

Now, substitute this simplified expression back into the differential equation:

dy/dx = (x^2 + 4x - 4y + 4) / (x^2 - 4)

This is a first-order linear homogeneous differential equation. To solve it, we can use the integrating factor method.

First, let's write the equation in the standard form: dy/dx + P(x)y = Q(x)

dy/dx + (4x / (x^2 - 4))y = (x^2 + 4x + 4) / (x^2 - 4)

The integrating factor is given by the exponential of the integral of P(x):

μ(x) = exp ∫ (4x / (x^2 - 4)) dx

To find the integral, we can use substitution. Let u = x^2 - 4, then du = 2x dx:

μ(x) = exp ∫ (2x dx) / (x^2 - 4)

= exp ∫ (du / u)

= exp(ln|u|)

= |u|

Substituting back u = x^2 - 4:

μ(x) = |x^2 - 4|

Now, multiply the entire differential equation by the integrating factor:

|x^2 - 4| dy/dx + (4x / (x^2 - 4)) |x^2 - 4|y = (x^2 + 4x + 4) |x^2 - 4| / (x^2 - 4)

The left side can be simplified using the product rule for differentiation:

d/dx [ |x^2 - 4|y ] = (x^2 + 4x + 4) |x^2 - 4| / (x^2 - 4)

Now, integrate both sides with respect to x:

∫ d/dx [ |x^2 - 4|y ] dx = ∫ (x^2 + 4x + 4) |x^2 - 4| / (x^2 - 4) dx

Integrating the left side gives:

|x^2 - 4|y = ∫ (x^2 + 4x + 4) dx

= (1/3) x^3 + 2x^2 + 4x + C1

where C1 is the constant of integration.

Finally, divide both sides by |x^2 - 4| to solve for y:

y = [(1/3) x^3 + 2x^2 + 4x + C1] / |x^2 - 4|

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Answer: £67.875

Step-by-step explanation:

€90.50 x 3/4 = £67.875

A type II error would occur if the candidate is actually a drug user, but the drug test incorrectly classifies them as a nonuser.

A type II error occurs when the null hypothesis is not rejected, even though it is false.

In this case, the null hypothesis is that the job candidate is not a drug user.

Therefore, a type II error would occur if the candidate is actually a drug user, but the drug test incorrectly classifies them as a nonuser.

In other words, a type II error would occur if the drug test fails to detect drug use in a job candidate who is actually a drug user.

This means that the organization would mistakenly hire a drug user, which could have negative consequences for the workplace and potentially put the safety of others at risk.

To minimize the risk of type II errors, organizations should use drug tests that are as accurate as possible and consider using multiple types of tests or follow-up testing to confirm results.

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The graph of the equation y = -4 + 4 cos²θ represents a cardioid, which is a heart-shaped curve.

To convert the polar equation r = -4 sin θ to rectangular form, we can use the following relationships between polar and rectangular coordinates:

x = r cos θ

y = r sinθ

Substituting the given polar equation into these equations, we have:

x = -4 sinθ cosθ and y = -4 sinθ sinθ

x= -4 sinθ cosθ and y = -4 sin²θ

Since sin²θ = 1 - cos²θ, we can rewrite the equation for y as:

y = -4 (1 - cos²θ)

y = -4 + 4 cos²θ

Now we have the rectangular form of the equation, with x and y in terms of cosθ.

The graph of the equation y = -4 + 4 cos²θ represents a cardioid, which is a heart-shaped curve.

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[tex]\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta \pi r^2}{360} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=6\\ \theta =45 \end{cases}\implies A=\cfrac{(45)\pi (6)^2}{360} \\\\\\ A=\cfrac{9\pi }{2}\implies A=\cfrac{9(3.14) }{2}\implies A=14.13~yd^2[/tex]

When a point is reflected about the xy-plane, its z-coordinate is negated while its x and y-coordinates remain the same. the image of the point P(-2, 3, 5) after a reflection about the xy-plane is the point P'(-2, 3, -5).

what is coordinates ?

Coordinates are values used to indicate the position of a point in a coordinate system. A coordinate system is a system that uses one or more numbers, called coordinates, to determine the position of a point or object. For example, in a two-dimensional Cartesian coordinate system, a point is located by its distance from two perpendicular lines, the x-axis and the y-axis, which intersect at a point called the origin. The coordinates of a point are usually represented by an ordered pair (x, y) in this system. In a three-dimensional coordinate system, a point is located by its distance from three perpendicular planes, and its coordinates are usually represented by an ordered triple (x, y, z).

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The set of data that provides the clearest difference between the two samples is option (c): One sample has M = 20 with s2 = 10 and the other has M = 30 with s2 = 10. The reason is that the standard deviation is higher in both samples compared to the other options, and the means of the two samples are also relatively far apart, making the difference between the two samples clearer. I

n options (a) and (b), the means are not far enough apart to provide a clear difference, and in option (d), while the means are farther apart than in option (b), the standard deviation is the same, which makes the difference less clear.

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Answer: -8

Step-by-step explanation:

13x+22=x-34+5x

13x+22=6x-34 (combine like terms)

13x-6x+22=6x-6x-34 (subtract 6x on each side)

7x+22=-34

7x-22+22=-34-22 (subtract 22 on each side)

7x=-56

7x/7x = -56/7 (divide 7 on each side)

x=-8

Check the picture below.

Answer: C

Step-by-step explanation: Plug -2 in for x for each answer option and see which inequality correctly applies. If the "mouth" of the inequality is pointed to the right, that means that the answer when you plug in x is less than the answer value. If the mouth is pointed to the left, that means that when you plug in x, the answer is greater than the resulting value.

The Percentage error is 40%. This means that the recorded value is 40% lower than the true value. In other words, the recorded value is only 60% of the true value.

To find the percentage error, we need to calculate the difference between the recorded value and the true value, then divide that difference by the true value and multiply by 100 to get a percentage.

True value = 1,500

Recorded value = 900

Difference = True value - Recorded value

Difference = 1,500 - 900

Difference = 600

Percentage error = (Difference / True value) x 100

Percentage error = (600 / 1,500) x 100

Percentage error = 40

Therefore, the percentage error is 40%. This means that the recorded value is 40% lower than the true value. In other words, the recorded value is only 60% of the true value.

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Answer:

400

Step-by-step explanation:

V= [tex]\frac{1}{3}[/tex]Bh

the base is a square, so B= [tex]10^{2}[/tex] or 100

h= 12

B= 100

Bh= 1200

[tex]\frac{1200}{3}[/tex] = 400

V=400

To determine the number of different ratings combinations possible, we can use the combination formula. Since there are five possible ratings (very poor, poor, fair, good, and excellent) and 29 students in the class. Therefore, there are 46,376 different rating combinations possible for the 29 students in the statistics class.

The formula we can use is:
nCr = n! / r!(n-r)!
where n is the total number of items (in this case, the number of ratings), and r is the number of items we are choosing (in this case, the number of students).
Using this formula, we can find the number of different ratings combinations possible by plugging in the values:
nCr = 5! / 29!(5-29)!
nCr = 5! / 29!(-24)!
nCr = 5 x 4 x 3 x 2 x 1 / (29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
nCr = 657,800
Therefore, there are 657,800 different ratings combinations possible for the 29 students in the statistics class.
In this scenario, there are 29 students and a 5-point scale for rating their instructor. To determine the number of different rating combinations possible, we will use the concept of combinations with repetitions allowed.
In this case, the formula for combinations with repetitions is given by:
C(n+r-1, r) = C(n+r-1, n-1), where n is the number of ratings (5) and r is the number of students (29).
Using the formula, we get:
C(5+29-1, 29) = C(33, 29) = 33! / (29! * 4!)
Calculating the factorials and simplifying the expression, we get:
C(33, 29) = 46,376
Therefore, there are 46,376 different rating combinations possible for the 29 students in the statistics class.

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When you graph a quadratic function, its shape is called a parabola.

Option C is the correct answer.

We have,

A parabola is a curve that is formed when graphing a quadratic function.

It is a U-shaped curve that can open upwards or downwards, depending on the coefficients of the quadratic equation.

A quadratic function is a polynomial function of degree 2, which can be expressed in the form y = ax² + bx + c, where a, b, and c are constants and x represents the variable.

The coefficient a determines whether the parabola opens upwards or downwards.

If coefficient a is positive, the parabola opens upwards, and if coefficient a is negative, the parabola opens downwards.

The vertex of the parabola is the highest or lowest point on the curve, depending on the orientation.

Thus,

A parabola is a specific shape that is formed when graphing a quadratic function, characterized by its U-shape.

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according to question the surface integral is (32π - 192)/15.

To evaluate the surface integral, we need to parameterize the surface and find the surface element ds.

Let's consider the parameterization:

x = y^2 + z^2

y = y

z = z

The surface element can be found as:

ds = √(1 + (dx/dy)^2 + (dx/dz)^2) dy dz

ds = √(1 + 4y^2) dy dz

Now, we can rewrite the integral as:

∫∫s z^2 ds = ∫∫R (y^2 + z^2)^2 √(1 + 4y^2) dy dz

where R is the projection of the surface S onto the yz-plane, which is the region 0 ≤ y ≤ 2, -√(4 - y^2) ≤ z ≤ √(4 - y^2).

Let's evaluate the integral:

∫∫s z^2 ds = ∫0^2 ∫-√(4-y^2)^√(4-y^2) (y^2 + z^2)^2 √(1 + 4y^2) dz dy

Using cylindrical coordinates, we can rewrite the integral as:

∫0^2 ∫0^π/2 ∫0^2r (r^2 cos^2θ + r^2 sin^2θ)^2 r √(1 + 4r^2 sin^2θ) dr dθ dy

Simplifying and solving the integral, we get:

∫∫s z^2 ds = (32π - 192)/15

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Answer:

Scale factor is 2

Step-by-step explanation:

Not sure how to explain, just count the number of points the dilated figure is from the point of dillations compared to the original figure. 2/1=2.

The equation P (1 + r/100)n, wherein P = the principal, r = yearly rate of interest, and n = the number of years and time periods, yields the amount of money when interest is made worse annually.

Compound interest refers to the interest added to a loan or deposit. It is the concept that we use every day the most regularly. For an amount, compound interest is computed using either the principal and accrued interest. This is the primary distinction among compound and simple interest. The formula P (1 + r/100)n, where P = principal, r = yearly rate of interest, and n = the number of years or time periods, yields the amount if interest is compounded annually.

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Answer:

3/28

Step-by-step explanation:

5 + 3 = marbles in total.

P(first blue) = 3/8

P(second blue) = (3-1) / (8-1)  = 2/7

P(selecting 2 blue) = (3/8) X (2/7) = 6/56 = 3/28

Answer:

0.09 l

Step-by-step explanation:

Given:

900 ml ⇒ 10s

Required:

Liters per second

Analyze:

Let the unknown amount be x.

900 ml ⇒ 10s

         x ⇒ 1s

Solve:

Use cross multiplication

900 × 1 = 10 × x

    900 = 10x

Divide both sides by 10.

90 ml = x

To covert the answer into liters divide it by 1000.

0.09 l = x

Paraphrase:

We breathe 0.09 liters of air per second

Answer:

(a) $ 7000

(b) $ 5600

Step-by-step explanation:

discount = 20%

20% = $1400

1% = 1400/20

original price = 100%

= 1400/20 × 100

= $7000

sale price = 80%

= 1400/20 × 80

= $5600

The probability that the sample mean will be between 295 and 305 can be determined using the Central Limit Theorem and the properties of the normal distribution.

According to the Central Limit Theorem, for a large sample size (n ≥ 30), the distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. In this case, since the sample size is 64, we can assume that the sample mean will follow a normal distribution.

To find the probability that the sample mean will be between 295 and 305, we need to standardize the sample mean using the formula z = (x - μ) / (σ / sqrt(n)), where x is the given range (295 to 305), μ is the population mean (300), σ is the population standard deviation (12), and n is the sample size (64).

By calculating the z-scores for the lower and upper limits of the range and referring to the standard normal distribution table, we can find the corresponding probabilities. The probability can be calculated by subtracting the cumulative probability corresponding to the lower limit from the cumulative probability corresponding to the upper limit.

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